Example 1.  Use the method of "data linearization" to find the logistic curve that fits the data for the population of the U.S. for the years 1900-1990.  Fit the curve    to the census data for the population of the U.S.

 Date Populatlion 76094000 92407000 106461000 123076741 132122446 152271417 180671158 205052174 227224681 249464396

Solution 1.

Enter the data points into a two dimensional array using millions.  Be careful with your typing !

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Next, a limiting population L, or "carrying capacity" must be estimated.  For this data the number L is not too sensitive, but must be larger than the largest ordinate so that the values    are not complex numbers. For illustration, we choose  L = 800  million.

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Do a series of intermediate computations.

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Now glue together the transformed parts to form the pairs  .

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Now use the Mathematica procedure  Fit  to get the least squares line in XY-space.  Then we shall graph this line in the transformed XY-plane.

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Plot the least squares line in XY-space.

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So the coefficients  A  and  B  are located at nodes  (2,1)  and  (1), respectively:

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Use    and  a = A  to get the coefficients of     back in the original  xy-plane.

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When we form the function, we must adjust "x" because we shifted the abscissas to the left.  The actual form of the answer is a little different than what we original planned.

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Now graph the function  .

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Remark.  The data for this example can be obtained from the  U.S. Census Bureau, Historical National Population Estimates:  July 1, 1900 to July 1, 1999.

(c) John H. Mathews 2004